I have a miner that I boarded stuck at an NPC shipyard. I don't remember if I ordered it to be repaired or just get some drones but it's been stuck there for a long time. One of the 2 L ship traders on that shipyard is showing as busy (can't buy ships through them) while the other seems fine, likely the busy one is busy with my ship. I can see my ship on the property window but cannot order it to do anything (other than joining my squad but that doesn't help with anything).

How do I get the ship unstock? Is there a way to get more information on what is going on (like what exactly is the problem, I imagine there's some sort of resource shortage although I checked and everything seems fine but I can double check)?

1) Contact the ship dealer on that shipyard responsible for relevant ship type (L/XL). If the ship is still undergoing some procedure (repair, upgrade, etc.), there would be a dialog option "Show Resource Status", which will open the resource page.

1.1) If there is a shortage, you can deliver required wares yourself using an appropriate freighter and corresponding trade offer for that shipyard, or wait until an NPC does so.

2) Sometimes even when resources are OK and the operation is done, the ship may get stuck for a while. If you're still in the same zone, try leaving it, i.e. flying into Empty Space or another zone, and wait a bit. After you hear the ship dealer contact you and tell you that "Ship is ready", wait another 30 seconds or so for the ship to undock and all ship commands to become available again (as some will be grayed out while it's undocking).

EDIT: Oh, also, since you boarded the ship... if you've ordered drones at the shipyard, it may take a long while until this can be completed, if you've destroyed the drone bay during boarding, because that bay needs to be repaired first. To remotely check whether it's intact or not, open the ship info page, and look at the numbers next to Units. If it says something like 0/0, or 10/0, or whatever (the second 0 is the relevant one) - that means the bay is still not repaired. When it is repaired, that second number will show the actual drone capacity instead of 0.

1) Contact the ship dealer on that shipyard responsible for relevant ship type (L/XL). If the ship is still undergoing some procedure (repair, upgrade, etc.), there would be a dialog option "Show Resource Status", which will open the resource page.

2) Sometimes even when resources are OK and the operation is done, the ship may get stuck for a while. If you're still in the same zone, try leaving it, i.e. flying into Empty Space or another zone, and wait a bit. After you hear the ship dealer contact you and tell you that "Ship is ready", wait another 30 seconds or so for the ship to undock and all ship commands to become available again (as some will be grayed out while it's undocking).

I've spent many hours outside of that system, following the main plot. Initially I forgot about the ship only to realize later that it's been sitting there for probably a whole game day.

EDIT: Oh, also, since you boarded the ship... if you've ordered drones at the shipyard, it may take a long while until this can be completed, if you've destroyed the drone bay during boarding, because that bay needs to be repaired first. To remotely check whether it's intact or not, open the ship info page, and look at the numbers next to Units. If it says something like 0/0, or 10/0, or whatever (the second 0 is the relevant one) - that means the bay is still not repaired. When it is repaired, that second number will show the actual drone capacity instead of 0.

Everything seems to have been repaired, see the above pictures. When I go into details for the Repair Engineer on that ship it says "Analyzing Components".

Hmm, these were all the things I could think of off the top of my head. If you're running vanilla, perhaps you could upload a save somewhere, maybe if I were to see it in game I'd recall something else?

I suggest you travel to Far Out, replace the captain on that miner, and fire that incompetent lady that doesn't even seem to know how to undock
(personal recommendation: keep her in the bay until the climax of the plot, and let her off there, that should teach her how to navigate to save her life, literally )

It's weird though, I think the only time I saw that particular behavior was back with the Taranis they have you capture during the plot, but then again, that one refused to move at all until I kicked out its old captain. Your Fedhelm had to move at least to get to that cradle though?..